Unit 3 of 10
Real and functional analysis
Fell from 9 questions to 3 in 2026. With only two full papers that is a range, not a trend: plan for anything from 3 to 9.
- 3–9questions expected next time
- 9asked in 2025
- 3asked in 2026
- 1idea asked in both papers
Do not drop it. Uniform convergence and multivariable limits are quick marks.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Uniform convergence of nx/(1 + n²x²)
Show answerHide answer
Its maximum is 1/2, at x = 1/n, so it is not uniform on any interval containing 0. On [a, ∞) with a > 0 it is uniform.
Asked in 2026
3 questions. Tap one for the answer and working.
Q33F_n to f uniformly, each f_n uniformly continuous on metric space X: what follows for fA
Uniform limit of uniformly continuous functions is uniformly continuous (3-epsilon argument), on any metric space.
Sure
Q44F_n(x)=nx/(1+n^3x^2)-type series on [0,inf): uniform convergence and interchange with integralD
f_n = nx/(1+n^2x^2): sum_n f_n(x) ~ sum 1/(nx) diverges for x>0, so the series is not uniformly convergent (not even pointwise); term-by-term integration over [0,inf) invalid (int f_n diverges, sup f_n=1/2 not -> 0).
Fairly sure
Two solves agree. B is arguable only in the sense that both sides are infinite.
Q60F of bounded variation and in L^2: strongest valid conclusion on Fourier coefficients a_nB
BV implies a_n = O(1/n); f in L2 gives sum |a_n|^2 < inf. A fails (sawtooth a_n ~ 1/n), C fails, D fails for discontinuous BV.
Sure
Asked in 2025
9 questions. Tap one for the answer and working.
Q38T self-adjoint iff ⟨Tx,x⟩ realC
T self-adjoint iff <Tx,x> real for all x (complex Hilbert space).
Matches the official key
Q39Order: l² Hilbert → Banach → normed → topologicalB
Implication chain: l2 Hilbert (II) => Banach (III) => normed (IV) => topological (I): II, III, IV, I. Reverse order (B) is the other reading.
Matches the official key
Official key (10.05.2025) says B: the order is weakest to strongest. My first solve said C (strongest to weakest).
Q40Xy/√(x²+y²): continuity/differentiability at originD
|f| <= r/2 so continuous; f_x(0,0)=f_y(0,0)=0; along y=x, f/r = 1/2 != 0 so not differentiable.
Matches the official key
Q41Interval of uniform convergence of a series of functionsD
On [1,inf), 1/(1+n^2 x) <= 1/n^2, Weierstrass M-test. On (0,c) not uniform (tail unbounded as x->0).
Matches the official key
Q42Ordering of series/convergence resultsC
Chain of implications: partial sums converge (II) => series converges (III) => a_n->0 (IV) => {a_n} convergent hence Cauchy (I).
Matches the official key
Q43Max of sin x sin y sin(x+y)B
Max of sin x sin y sin(x+y) is (sqrt3/2)^3 = 3sqrt3/8 at x=y=pi/3.
Matches the official key
Q44Divergence of a series ∑uₙ with reasonA
u_n = 1/((2/3)^n+1) -> 1 != 0, so series diverges; R is the divergence (nth term) test.
Matches the official key
Q45Domain of uniform convergenceD
f_n = n x e^{-n x^2}; sup on [0,a] is sqrt(n/2)e^{-1/2} -> inf (no uniform). On [1,2], f_n <= 2n e^{-n} -> 0, uniform. Exponent read as -n x^2.
Matches the official key
exponent in the printed f_n is small; read as -n x^2. If it were -x^2 the sequence would not converge pointwise at all, so D stands either way among the options.
Q79Max distance between curves on [0,1]C
max of x - x^2 on [0,1] at x=1/2 gives 1/4
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Riemann integrability criteria
- Improper integrals
- Lebesgue measure and integral (asked in 2019)
- Compactness and connectedness in metric spaces
- Hahn–Banach, open mapping, closed graph
- Operator norm (asked in 2019)
Full syllabus for this unit
Sequences and series of functions, uniform convergence, Fourier series, Riemann and improper integrals, bounded variation, Lebesgue measure and integral, several-variable calculus, metric spaces, normed and Banach spaces, Hahn–Banach, open mapping, closed graph, Hilbert spaces.